Date: Dec 13, 2012 5:50 PM
Author: Virgil
Subject: Re: On the infinite binary Tree

In article 
<a618812f-4c3c-4703-838c-7cafe1060c59@fn10g2000vbb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 13 Dez., 21:02, Zuhair <zaljo...@gmail.com> wrote:
> > On Dec 13, 3:56 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >

> > > No? Nearly every real number is undefinable. The measure of definable
> > > reals is 0. If most reals are non-definable, why must all reals of
> > > every Cantor list always be definable? If all reals of the list are
> > > definable, then they belong to a countable set. Then we cannot prove
> > > uncounatbility. Or can we prove that the set of definable reals is
> > > uncountable - because it is countable but there are, somewhere else,
> > > undefinable "reals"?

> >
> > > Regards, WM
> >
> > Cantor's list do contain non definable reals.


The more formal , and instructive, name for them is "inaccessible
numbers". They are the numbers that, to make the reals uncountable must
exist, but are beyond the power of mathematics to be individually
accessed. We cannot access more than countably many reals.

And since they cannot be accessed, they cannot appear in any lists.
>
> Which one in what line? What is the corresponding digit of the
> diagonal?
>

> > Actually some diagonals
> > of Cantor's are non definable.

>
> Lists containing undefinable entries do not supply diagonals at all.


But such a list would have to contain inaccessible numbers which means
they would have to be accessible after all. Contradiction.
>
> > The bijective function between all
> > definable reals and the set N of all naturals is NON definable set!


The problem is that every such list will define numbers not listed in it.
>
> That is completely without interest.


What is of no interset t WM is usually the very thing proving him wrong
again.
> >
> > Note: definable is short for "definable by parameter free finite
> > formula"

>
> No. Definable means "definable by a finite word". Everything else is
> "undefinable".


Accessible and inaccessible are related, but better, representations of
the relevant ideas.
>
> Regards, WM

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